ASITEilETIC THE SECOND class of numbers, so again Dedekind overcame his difficulties by taking refuge in che infinite. The concept of an ideal is not hard to grasp, although there is one twist - the more inclusive class divides the less inclusive, as will be explained in a moment - which shocks common sense. However, common sense was made to be shocked; had we nothing less dentable than shock-proof common sense we should be a race of mongoloid imbeciles. An ideal must do at least two things: it must leave common (rational) arithmetic substantially as it is, and it must force the recalcitrant alge- braic integers to obey that fundamental law of arithmetic - unique decomposition into primes - which they defy. The point about a more inclusive class dividing a less inclu- sive refers to the following phenomenon (and its generalization, as stated presently). Consider the fact that 2 divides 4 - arith • metically, that is, without remainder. Instead of this obvious fact, which leads nowhere if followed into algebraic number fields, we replace 2 by the class of all its integer multiples, ... , — 8, — 6, — 4, — 2,0,2,4,6,8, ... As a matter of convenience we denote this class by (2). In the same way (4) denotes the class of all integer multiples of 4. Some of the numbers in (4) are ... , — 16, —12, — 8, — 4, 0, 8, 12, 16, ... It is now obvious that (2) is the more inclusive class; hi fact (2) contains aU the numbers in (4) and in addition (to mention only two) — 6 and 6. The fact that (2) contains (4) is symbolized by writing (2)|(4). It can be seen quite easily that if m, n are any common whole numbers then (m) \ (n) when, and only when, m divides n. This might suggest that the notion of common arithmetical divisibility be replaced by that of class inclusion as just described. But this replacement would be futile if it failed to preserve the characteristic properties of arithmetical divisi- bility. That it does so preserve them can be seen in detail, but oae instance must suffice. If m divides n, and n divides J, then m divides I - for example, 12 divides 24 and 24 divides 72, and 12 does in fact divide 72. Transferred to classes, as above, this becomes: if (m) \ (n) and («) j (Z)» then (m) j (Z) or, in English, if the class (m) contains the class (n), and if the class (n) contains the class (I), then the class (m) contains the class (I) - which 577